9 THE PLANCHEREL-GODEMENT THEOREM 381 The Plancherel-Godement theorem applies in particular when the bitrace g is of the form (x, y) \-+f(y*x), where /is a positive linear form on A (and therefore a trace since A is commutative). When g comes from a trace/ the formula (15.9.2.1) leads us to ask whether we also have (15.9.3) = f JS A partial answer to this question is provided by the following theorem: (15.9.4) (Bochner-Godement theorem) (i) Let f be a positive linear form on a commutative algebra A with involution, such that the bitrace 0(x, y) = f(xy*) satisfies the hypotheses of (15.9.2). Then, if the formula (15.9.3) is true and if the measure mg is bounded, f satisfies the condition (B) There exists a real number M > 0 such that \f(x)\2 £ M -f(xx*)for all x e A. (ii) Conversely, let f be a positive linear form on A which satisfies the condition (B), and suppose that the corresponding bitrace g(x, y) — f(xy*) satisfies (U) and is such that the prehilbert space A/ng and the star algebra $0g are separable. Then g also satisfies (N), the measure mg is bounded, and the formula (15.9.3) is valid, (i) If mg is bounded and if (15.9.3) holds, then the Cauchy-Schwarz inequality (13.11.2.2) gives us f*dm/^ro,(S,)- f and hence the inequality of (B), with M = (ii) Recall that the definition of the Hilbert space Ug does not pre- suppose that the condition (N) is satisfied. The inequality in (B) can be put in the form \f(x)\2 ^ M\\ng(x)\\2, and shows that / vanishes on n^; hence we may write f=f'°ng, where /' is a linear form on A/ng. Also \f'(n9(x))\2 = M ||7T0(x)||2, so that/' is continuous on the prehilbert space A/110 (5.5.1) and therefore extends to a continuous linear form on the Hilbert space H^ (5.5.4). Hence, by (6.4.2), there exists a well-determined vector a e H^ such that (15.9.4.1) f(x) = (n,(x)\d). From this it follows that, for all # e A, we have (15.9.4.2) V9(x)-a = n,(x). \x(y)$(l) dm(%). The argument of